A163190 a(n) = Sum_{k=0..n} C(n,k)*sigma(n,k) for n>0 with a(0)=1.

1, 2, 13, 72, 722, 7808, 122538, 2097280, 43444163, 1000262656, 25997950850, 743008372736, 23312187863060, 793714773262336, 29197324076701082, 1152921975865606144, 48663045048486723204, 2185911559738696663040

Offset

0,2

Comments

Definition: sigma(n,k) = sigma_k(n) = Sum_{d|n} d^k.

Links

Table of n, a(n) for n=0..17.

Formula

a(n) = Sum_{d|n} (1+d)^n for n>0 with a(0)=1.

a(n) ~ exp(1) * n^n. - Vaclav Kotesovec, Oct 08 2016

Mathematica

Flatten[{1, Table[Sum[Binomial[n, k] * DivisorSigma[k, n], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Oct 08 2016 *)
a[n_] := DivisorSum[n, (1+#)^n &]; a[0] = 1; Array[a, 18, 0] (* Amiram Eldar, Aug 29 2023 *)

Prog

(PARI) {a(n)=if(n==0, 1, sumdiv(n, d, (1+d)^n))}
(PARI) {a(n)=if(n==0, 1, sum(k=0, n, binomial(n, k)*sigma(n, k)))}
(Python)
from sympy import divisors
def A163190(n): return sum((1+d)**n for d in divisors(n, generator=True)) if n else 1 # Chai Wah Wu, Nov 21 2023

Crossrefs

Cf. A000203 (sigma), A007318 (binomial).

Sequence in context: A097349 A289790 A109112 * A242991 A240549 A004027

Adjacent sequences: A163187 A163188 A163189 * A163191 A163192 A163193

Keyword

nonn

Author

Paul D. Hanna, Jul 22 2009

Status

approved